It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms. In this version, the Jones polynomial can be generalized fromS
3 to arbitrary three manifolds, giving invariants of three manifolds that are computable from a surgery presentation. These results shed a surprising new light on conformal field theory in 1+1 dimensions.

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Quantum field theory and the Jones polynomial

The generalization of Jones polynomial of links to the case of graphs inR
3 is presented. It is constructed as the functor from the category of graphs to the category of representations of the quantum groups.

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Ribbon graphs and their invariants derived from quantum groups

We present a new construction of the quantum Hubert space of ChernSimons gauge theory using methods which are natural from the threedimensional point of view. To show that the quantum Hubert space associated to a Riemann surface Σ is independent of the choice of complex structure on Σ, we construct a natural projectively flat connection on the quantum Hubert bundle over Teichmuller space. This connection has been previously constructed in the context of two-dimensional conformal field theory where it is interpreted as the stress energy tensor. Our construction thus gives a (2 + 1 )-dimensional derivation of the basic properties of (1 + 1)-dimensional current algebra. To construct the connection we show generally that for affine symplectic quotients the natural projectively flat connection on the quantum Hubert bundle may be expressed purely in terms of the intrinsic Kahler geometry of the quotient and the Quillen connection on a certain determinant line bundle. The proof of most of the properties of the connection we construct follows surprisingly simply from the index theorem identities for the curvature of the Quillen connection. As an example, we treat the case when Σ has genus one explicitly. We also make some preliminary comments concerning the Hubert space structure.

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Geometric quantization of Chern-Simons gauge theory

We study a system of functional relations among a commuting family of row-to-row transfer matrices in solvable lattice models. The role of exact sequences of the finite-dimensional quantum group modules is clarified. We find a curious phenomenon where the solutions of those functional relations also solve the so-called thermodynamic Bethe ansatz equations in the high temperature limit for sl(r+1) models. Based on this observation, we propose possible functional relations for models associated with all the simple Lie algebras. We show that these functional relations certainly fulfill strong constraints coming from the fusion procedure analysis. The application to the calculations of physical quantities will be presented in the subsequent paper.

Functional relations in solvable lattice models i: functional relations and representation theory

Correlation functions of the XXZ Heisenberg spin- chain in a magnetic field

Background of the problem The spin $1/2$ XXZ model for $\Delta <-1$ The six-vertex model in the anti-ferroelectric regime Solvability and symmetry Correlation functions-physical derivation Level one modules and bosonization Vertex operators Space of states-mathematical picture Traces of vertex operators Correlation functions and form factors The $XXX$ limit $q\rightarrow-1$ Discussions List of formulas

Algebraic analysis of solvable lattice models

Preface 1. The KdV equation and its symmetries 2. The KdV hierarchy 3. The Hirota equation and vertex operators 4. The calculus of Fermions 5. The Boson-Fermion correspondence 6. Transformation groups and tau functions 7. The transformation group of the KdV equation 8. Finite dimensional Grassmannians and Plucker relations 9. Infinite dimensional Grassmannians 10. The bilinear identity revisited Solutions to exercises Bibliography Index.

Solitons : differential equations, symmetries and infinite dimensional algebras

This volume will be the first reference book devoted specially to the Yang-Baxter equation. The subject relates to broad areas including solvable models in statistical mechanics, factorized S matrices, quantum inverse scattering method, quantum groups, knot theory and conformal field theory. The articles assembled here cover major works from the pioneering papers to classical Yang-Baxter equation, its quantization, variety of solutions, constructions and recent generalizations to higher genus solutions.

Yang-Baxter equation in integrable systems

Conformal field theory and solvable lattice models

Selected papers presented at two symposia at the Research Institute for Mathematical Sciences in Kyoto and at Kyuzeso in Katata in October 1988. Annotation copyright Book News, Inc. Portland, Or.

道夫神保

Papers

Algebraic analysis of solvable lattice models

Solitons : differential equations, symmetries and infinite dimensional algebras

Yang-Baxter equation in integrable systems

Conformal field theory and solvable lattice models

Integrable systems in quantum field theory and statistical mechanics

道夫 神保

Papers

Algebraic analysis of solvable lattice models

Solitons : differential equations, symmetries and infinite dimensional algebras

Yang-Baxter equation in integrable systems

Conformal field theory and solvable lattice models

Integrable systems in quantum field theory and statistical mechanics

道夫神保